Weakly proper moduli stacks of curves

TL;DR

This paper introduces the concept of weakly proper algebraic stacks to characterize certain non-separated moduli problems and demonstrates their weak properness for stacks of curves with various singularities, aiding the log minimal model program.

Contribution

It defines weakly proper stacks and develops techniques to prove their properness without GIT semistability analysis, applied to moduli of curves with singularities.

Findings

Defined weakly proper algebraic stacks.

Proved weak properness of stacks with nodes, cusps, tacnodes, and ramphoid cusps.

Established groundwork for constructing projective moduli spaces as log canonical models.

Abstract

This is the first in a projected series of three papers in which we construct the second flip in the log minimal model program for $\bar{M}_g$. We introduce the notion of a weakly proper algebraic stack, which may be considered as an abstract characterization of those mildly non-separated moduli problems encountered in the context of Geometric Invariant Theory (GIT), and develop techniques for proving that a stack is weakly proper without the usual semistability analysis of GIT. We define a sequence of moduli stacks of curves involving nodes, cusps, tacnodes, and ramphoid cusps, and use the aforementioned techniques to show that these stacks are weakly proper. This will be the key ingredient in forthcoming work, in which we will prove that these moduli stacks have projective good moduli spaces which are log canonical models for $\bar{M}_g$.